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What is an example of a Poisson distributed random variable with parameter $\lambda > 0$?

gt6989b
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pba
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1 Answers1

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From Wikipedia's entry on the Poisson distribution:

For instance, a call center receives an average of 180 calls per hour, 24 hours a day. The calls are independent; receiving one does not change the probability of when the next one will arrive. The number of calls received during any minute has a Poisson probability distribution with mean 3: the most likely numbers are 2 and 3 but 1 and 4 are also likely and there is a small probability of it being as low as zero and a very small probability it could be 10. Another example is the number of decay events that occur from a radioactive source during a defined observation period.

UPDATE

I don't quite understand what a more mathematical example would be. Perhaps if you have a discrete random variables $X,Y,Z$ with the pmfs $$ \begin{split} f_X(k) &= \frac{1/e}{k!}, \quad k \ge 0 \\ g_Y(k) &= \frac{e^{k-e}}{k!}, \quad k \ge 0 \\ h_Z(k) &= \frac{1}{e^\pi} \frac{\pi^k}{k!}, \quad k \ge 0 \end{split} $$ then $X \sim \mathcal{P}(1), Y \sim \mathcal{P}(e)$ and $Z \sim \mathcal{P}(\pi)$.

gt6989b
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  • Is there also a more mathematical example, except the real-life examples for function X? – pba Oct 13 '22 at 19:33
  • @pba not sure what that would mean, but please see the update. – gt6989b Oct 14 '22 at 06:45
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    The "more mathematical" way the Poisson($\lambda$) distribution appears is as the $n \to \infty,p=\lambda/n$ limit of the binomial distribution. It is worth thinking about how this compares to the Poisson process. – Ian Oct 14 '22 at 07:21
  • For a mathematical way, I was referring to a concrete formula for the random variable X. – pba Oct 14 '22 at 08:58
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    If Ian's hint isn't concrete enough perhaps the following is ? Draw independent exponentially distributed waiting times $\sigma_i$ (these are the times between two calls to the center). $\tau_n=\sigma_1+...+\sigma_n$ is the time of the $n$-th call. The number of calls that have arrived within, say, one day will be Poisson distributed. With the link I have given you can even simulate this. – Kurt G. Oct 14 '22 at 13:00
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    There's always a "concrete formula" for any 1D random variable as $Q(U)$ where $Q$ is the quantile function and $U$ is uniform($0,1$) distributed. – Ian Oct 14 '22 at 20:29